When i start laptop after it has been off for a while (more than few minutes), the screen either remains completely dark (off) or white (with borders darker). After it has been on for a while, restarting it makes screen work normally. If screen is working fine, restarting laptop never results in screen problems. As long as i use it as desktop replacement (never turn off laptop), it works fine. Turning off just screen (after it has been idle for some time), never causes any problem (moving mouse or pressing any key immediately brings back screen).

It seems as if laptop has to warm up a little before screen can work, does anyone has any clue what could be the underlying issue ?

Screen either remains OFF (not dark/black) or become WHITE. It has nothing to do with HDD, i can connect and use external monitor without any issue. Anyway, I upgraded to SSD a while back and there was no change. I have sometimes used USB boot to boot into Ubuntu, i had the same issue for all these scenarios. I think some component has to get warmed or charged for screen to turn on but i don't know what that component could be.

If i start laptop after it has been OFF for long time, screen almost always remains OFF, if i reboot it within few minutes, screen often becomes WHITE. If I reboot after 5-10 min, screen works properly.

I would try checking to see how much an inverter for the LCD screen would cost (if you're not afraid to pop the trim off of the screen that is). I know my thinkpad's pretty simple and it's a few screws and pop tabs around the outside edge and the power supply/inverter is usually on the bottom of the panel under the actual LCD screen panel, it's usually a connector coming from the laptop body itself, and the other jack runs up to the LCD panel itself. Very easy to change and not too crazy expensive, but it sounds like it's not working properly and not sending the right voltage which over time could end up damaging the panel.

I know I only paid about 10-15 bucks for one off of ebay, and if it's a super old laptop might not be worth trying, but if it's a newer one that's past the warranty you might want to try it. If you want I could take apart and show you some reference photos from my older thinkpad that's got a bad mobo, which is what it ended up being on mine, not the inverter. That's unfortunately one of the other parts that could be bad, mine the lcd wouldn't work all the time either and I could do out the vga port, but my laptop also started randomly shutting down, and if yours isn't doing that yet, then I'd try the inverter first.